Find the height of the following rectangle

Question

We have a rectangle ABCD, and a point P on the diagonal AC. From P we see BC at an angle $\alpha$. Knowing $\alpha$, AP and AB, find BC.

Here's a probably unnecessary illustration of the problem, with known data in red:

I am lost with this, certainly, extremely simple problem. I might use the cosine law to define BC as a function of $\alpha$, PC and PB, but I find no way of finding those latter. I thought of finding PB constructing a parallelogram including sides PB and AP, having BC as diagonal, but I'm stuck there since I, in my poor knowledge on solving parallelograms, I would need the other diagonal of the parallelogram to find PB. On the other hand, I looked where could I transpose $\alpha$ as to express its trigonometric functions using known information, but didn't found anything useful.

Sorry for coming to you with high-school problems (I feel ridiculous to be asking this after three semesters of real analysis, one of complex, and one of linear algebra). I'd appreciate, in addition to a method of resolution, ressources that might help me get a comprehensive approach to planar, high-school like geometry.

Answer 1

1. Find $\angle APB = 180º - \alpha$.
2. Use the sine rule in $\Delta APB$ to find $\angle PBA$.
3. Find $\angle PAB = 180º - \angle APB - \angle PBA$.

then since $\Delta CAB$ is right-angled, $\tan \angle PAB = \frac{BC}{AB}$, so you now have $BC$.